WASHINGTON — The stalemate in Washington over immigration policy is producing collateral damage in New England: thousands of unfilled jobs.

Federal rules allowing foreign workers to come to the United States for high-skilled jobs are caught in the battle between Republicans and Democrats over border security and giving illegal immigrants a pathway to citizenship. As a result, the region’s business leaders say, there are not enough skilled workers to fill essential jobs.

“The industry in Boston is very innovation-driven, and technical talent is critical to fuel that innovation,” said Jeffrey Bussgang, a Boston venture capitalist who teaches entrepreneurship at Harvard Business School, where he said foreign students make up about a quarter of each class. “We spend millions of dollars marketing to foreign students to come to our world-class universities, and then when they are ready to start companies here, we kick them out. It’s absurd.”

With Congress disbanded for its August recess without agreeing on short- or long-term plans for immigration, New England business leaders say they are frustrated that their top priority of increasing the number of visas for skilled foreign workers has been shunted aside in the partisan feud.

Intel, with 1,400 workers in Hudson, is lobbying the Obama administration to take executive action on several fronts, including reducing the green card backlog so that qualified candidates do not have to wait years for permanent residency.

Intel submits between 800 and 900 applications nationally for so-called H-1B visas each year for US-educated foreign-born workers, and about three-quarters are granted, said Peter Muller, Intel’s director of immigration policy who is based in Washington.

“It’s something we rely on to be able to fill jobs that we are unable to fill otherwise,” Muller said.

US Representative Michael E. Capuano, a Massachusetts Democrat, called the current immigration policy preventing foreign graduates of American universities from working in the country “absolutely insane,” according to the State House News Service.

“If we don’t keep them, they will go home and they will open the businesses to compete against us,” Capuano said at an event at Google’s Cambridge office, State House News said.

Capuano has sponsored a bill that would make it easier for foreigners who have US doctorate degrees in science, technology, engineering, and math to work in the United States.

If Congress does not take action, Bussgang said, then states must. He supports an economic development bill recently proposed by Governor Deval Patrick that would get around federal limitations and allow foreign students who want to stay in Massachusetts after college or graduate school to do so through an Entrepreneurship in Residence program. The graduates would work at local universities part time while developing start-ups.

Thomas Ketchell, a 25-year-old entrepreneur from Belgium whose fledging education company, Hstry, is based in the Back Bay, said the difficulty of securing an H-1B visa for himself and his English, French, Dutch, Canadian, and Chinese colleagues is costing the company thousands of dollars in legal fees and lost productivity because members of the team must leave the country every three months when their tourist visas expire.

Only two of the company’s four founders are able to remain in Boston this summer; the others had to go back to their home countries due to visa issues. When Ketchell returned to Boston in May, he was detained at Logan Airport and questioned for two hours by immigration officials who suspected him of working without a work permit. He said he was told to “watch out and get my paperwork in order.”

He considered himself lucky because he was allowed to stay for three months.

“It’s a real shame because there are so many entrepreneurs who actually want to come here to create jobs for Americans who are not able to do so,” Ketchell said.

Business groups want to increase the number of H-1B work visas from the current 85,000 limit to as many as 180,000 a year, as a comprehensive immigration bill passed by the Senate last year would have allowed. The catch-all immigration reform measure was never taken up in the House.

New England business leaders are pushing for a stand-alone bill to raise the number of visas issued to foreign workers.

“Both the House and Senate and both parties would say that’s a no-brainer, because it’s important to the economy,” said James Brett, president and chief executive of the New England Council, an alliance of universities, hospitals, and corporations. “Unfortunately, there are members of Congress who saywe need everything in the immigration bill. But I’m of the Ted Kennedy school of politics. A half a loaf is better than no loaf.”

When the latest H-1B visa lottery was announced in April, all 85,000 visas — the 65,000 allowed under the cap plus an additional 20,000 for workers with advanced degrees — were gone within five days. (During the program’s peak between 2001 and 2003, Congress raised the annual cap to 195,000 visas, but the number was never reached.)

In 2013, the US Citizenship and Immigration Services agency approved 10,882 H-1B visas for Massachusetts, according to the latest data available. Massachusetts ranks seventh in the number of such visas approved, behind California, New Jersey, Texas, New York, Illinois, and Maryland.

A national business coalition called Partnership for a New American Economy, launched in part by former New York mayor Michael Bloomberg and backed by Boston’s technology companies, universities, and hospitals, released a report in June highlighting how the tech industry and the country’s economy could have grown faster amid the recession if a larger number of foreign engineers and programmers were granted work visas.

Critics, however, say that expanding the number of H-1B visas is simply a way for businesses to hire workers at a lower wage than what US workers would be paid. It is a myth that companies only hire foreign workers after considering the American talent pool, said Ron Hira, a professor of public policy at Rochester Institute of Technology.

“H-1B workers can legally be paid below market wages,” said Hira, who also testified before the Senate Judiciary Committee on the issue last spring. “The majority of these workers have ordinary skills and capabilities that are readily obtained in the US labor market.”

Some business coalitions have reservations about the idea of carving out a narrow visa bill instead of making it part of a broader immigration measure.

“You can’t just peel off your own little piece,” said Joe Green, president of FWD.us, an immigration reform advocacy group of technology companies founded by Facebook’s chief executive, Mark Zuckerberg. Green hopes that House Republicans will revisit the issue after the November midterm elections as part of a comprehensive package.

“For the Republicans, the politics become even more stark after the midterms looking ahead to the presidential election,” he said. “If they don’t improve their performance among Asians and Latinos, it just makes it more difficult for them to win.”

The Greater Boston Chamber of Commerce, which spearheaded a national coalition last year called “Business for Skilled Worker Immigration,” is calling upon business leaders in Republican states to step up the pressure on their congressional delegations for a deal, said Jim Klocke, executive vice president of the Greater Boston chamber.

“The issue playing out on the Texas-Mexico border is dominating every day,” he said. “We and others just need to continue stressing the importance of the visa issue.”